Bug? Logged out after clicking 'edit post'

Only in your site. Have you tried:

– deactivating all plugins to see if this resolves the problem. If this works, re-activate the plugins one by one until you find the problematic plugin(s).

– switching to the default theme to rule out any theme-specific problems.

– resetting the plugins folder by FTP or PhpMyAdmin. Sometimes, an apparently inactive plugin can still cause problems.

– re-uploading all files & folders – except the wp-content folder and wp-config.php & root .htaccess files – from a fresh download of WordPress. Make sure that you delete the old copies of files & folder before uploading the new ones.

I should have mentioned in my original post that I first experienced this problem when dealing with my first time password protected post.
That being said, I followed your first two pointers: plugin deactivation/reactivation and switching to default theme. The problem seem to have been solved. But then I quit the browser and found the following behavior:
Regardless of browser, the problem always occurs the first time I open the browser. Then it doesn’t reoccur until the next time I quit the browser. It almost sounds like a cookie related problem.